# I will set my face against that man

"I will be against that man" or "I will oppose that man". When the captives from Jerusalem arrive in Babylon, it will be obvious why God has punished them.

# set my face against

"stare angrily at"

# make him a sign and a proverb

"I will make that man to be a warning and a proverb"

# I will cut him off from the midst of my people

"I will cause him no longer to belong to my people"

# know that I am Yahweh

"understand that I am Yahweh, the one true God". Yahweh is implying that they will know that he is the one true God who has supreme authority and power. See Ezekiel 6:7.